Report on a problem studied at the Canadian Math-in-Medicine Study Group Toronto 2009
< http://www.maths-in-medicine.org/canada/2009/calcium-carbonate/ >
Proceedings of the OCCAM–Fields–MITACS Biomedical Problem Solving Workshop, 2009
Control of Calcium Carbonate Crystallization by a Serum
Protein
Problem Presenter: D. Bassett and J. Barralet (Strathcona Anatomy & Dentistry,
McGill University)
Contributors: C.S. Bohun (UOIT), C.J.W. Breward (University of Oxford), H. Huang
(York University), M. Morfin (University of Toronto), N. Nigam (McGill University), J.
Phillips (McGill University), B. Tilley (Franklin W. Olin College of Engineering), M.J.
Tindall (University of Reading), J.A.D. Wattis (University of Nottingham)
Report prepared by1 : Mario Morfin, C. Sean Bohun, and Jonathan Wattis
Abstract. Calcium carbonate crystals occur in a variety of shapes, each morphology having differing physicochemical properties. This problem involves the formation of calcium carbonate crystals in a gas diffusion process. Experiments showed
the formation of clear shapes in the presence of a serum protein. These structures
appear to be formed from calcium carbonate fibers arranged in cones, fiber bundles,
discs, and other shapes. Some characteristics of the crystals, such as the layering
of self similar structures, suggest a process which regulates crystallization. Here
we propose a mechanism for the formation of calcium carbonate fibers and their
assembly in the complex structures. We describe this process in two systems of
partial differential equations. We aim to simulate the growth of these crystals in
order to understand the effect of the concentrations and diffusion of the different
elements and compounds that are present in the reaction, in the global regulator of
the crystallization process.
1
[email protected], [email protected], [email protected]
c
2009
39
40
Control of Calcium Carbonate Crystallization by a Serum Protein
1 Introduction
Calcium carbonate, CaCO3 , is one of the three most common biominerals, the other two
being calcium phosphate and silica. When crystallised, CaCO3 forms diverse morphologies
with different mechanical, electrical and optical characteristics. This variety of morphologies makes calcium carbonate useful in industry. When biomineralization occurs, various
proteins are found in mineralized tissues such as bone, teeth and shells. The presence of
these proteins in the biomineralization process is known to effect the characteristics of the
final crystal.
The problem posed is to describe the crystallization of calcium carbonate in a gas
diffusion process. This process, which will be described in detail below, mimics the biomineralization of CaCO3 . It relies on the decomposition of ammonium carbonate to produce
carbon dioxide and ammonia gas in a closed environment. The crystallization is achieved
via the diffusion of carbon dioxide gas into a solution filled with calcium chloride.
It is well known that, in the absence of other agents, calcium carbonate will crystallize
into its most stable form: rhombohedral microcrystals of calcite, see Figure 1 (left). In
the experiments presented to the group, an external compound, a serum protein which we
denote by P , was introduced to the solution of calcium carbonate. It is known that the
serum protein, which is highly phosphorylated, is similar to those found in bone, serum
and milk, and inhibits the formation of large calcium carbonate crystals. The aim of the
experiment is to explore the effect of the serum protein on the crystallization process.
In the presence of serum protein under the conditions of our experiments, aggregate
structures, such as fibres, cones, plate-like floating shapes and fibre bundles, were observed.
These structures were formed as calcium carbonate fibre crystals bound together.
The crystals were found in the following configuration: the aggregate structures were
formed in the aqueous solution, attached to the surface of the liquid. The cone tips and the
binding point of the bundle were also attached to the surface, while the rest of the structure
was suspended in the solution. Calcite was formed attached to the bottom of the vessel.
Nanofibres were found loose at the bottom of the vessel after drying.
Figure 1 The image to the left shows rhombohedral calcite crystals. The presence of
serum protein changes the morphology as illustrated by the image to the right which
shows a beautiful set of crystallized cones.
Some global characteristics were observed. For example, in any one experiment, the
cone-like shapes appear to be similar, in that they had a similar angle at each apex, even
when embedded within each other. The lower boundary of the crystals, i.e. where the
crystal stops growing, appear to be constant along separate aggregate structures. (See
Control of Calcium Carbonate Crystallization by a Serum Protein
41
Figure 1, right picture and Figure 4, right picture.) This suggests the existence of a global
regulatory process.
In the present report, we propose a mechanism for the formation of calcium carbonate
fibres and crystals in the complex structures described above. Also, we present two systems of partial differential equations (pdes) which we use to simulate the growth of these
aggregate structures. From the simulations, we attempt to understand the impact of the
concentrations and diffusion of the various substances on this regulatory process.
2 The experimental gas-diffusion process
2.1 Methodology. A simple gas diffusion method was used to precipitate calcium
carbonate. It was carried out in a closed desiccator at room temperature. The serum
protein was dissolved in 100 mM CaCl2 aqueous solution and 1 ml of solution was placed
in the reaction vessel. The reaction vessel was then covered with parafilm and punctured
six times with a needle. Crushed ammonium carbonate was placed in a dish, covered with
parafilm, then punctured in the same way, placed in the bottom of the desiccator. Typically
the reaction was allowed to stand for three days, as this was deemed to be when the reaction
had reached completion; however, some samples were kept for shorter (one day) and longer
(twenty days) times. See Figure 2 for an illustration of the physical setup. Upon completion
of the crystallization, the calcium carbonate was removed, dried and examined by X-ray
diffraction, scanning and transmission electron microscopy (SEM and TEM).
Figure 2 Experimental apparatus for the gas diffusion process.
2.2 Chemical reactions. The ammonium carbonate sublimation chemical reactions
are
(NH4 )2 CO3(s) NH3(g) + CO2(g) + H2 O(g)
CO2(g) + H2 O(l) →
NH3(g) + H2 O(l) →
+
CO2−
3 + 2H
−
NH+
4 + OH
(2.1)
(2.2)
(2.3)
42
Control of Calcium Carbonate Crystallization by a Serum Protein
Carbon dioxide was released to the desiccator, and then dissolved in the solution forming
+
dissolved carbon dioxide as well as carbonate ions (CO2−
3 ) and H3 O . According to the
diffusion equation, under these experimental conditions, it should take around five hours
for these ions to diffuse to the bottom of the vessel. Carbonate ions then react, in the
presence of protein, with the Ca2+ ions to form CaCO3 in the aqueous solution, according
to the reactions
CaCl2 → Ca2+ + 2Cl−
Ca
2+
+ CO2−
3
CaCO3 .
(2.4)
(2.5)
The protein is about 200nm long, it has a molecular weight of about 60 kDa, and it is
highly phosphorylated. It is amphiphilic since its ends are hydrophobic whilst its centre is
slightly hydrophilic. If present at high enough concentration in water, it forms micelles. In
the presence of CaCO3 , in order to minimise energy, its ends will be in contact with CaCO3
rather than water, thus we might expect the polymer to interact with crystal surfaces once
the crystals grow to a large enough size.
2.3 Summary of experimental results. Two types of structures were observed after
the crystallization process: nanofibres and aggregated structures (the former appear to be
the building blocks of the latter). Nanofibres were between 1 and 2 μm wide, while the size
of the aggregated structures were hundreds of times larger (up to 0.5 mm).
At low concentrations of the serum protein (between 0.1 mg/ml and 0.5 mg/ml), only
nanofibres were observed. Cones and bundles of fibres were only found at protein concentrations above 1 mg/ml, see Figure 3 (right) and Figure 4 (left and right). At concentrations
lower than 0.1 mg/ml, the fibrous structure was lost — see the left panel of Figure 3.
Figure 3 These pictures show the effect of low concentration of protein. On the left,
the concentration is lower than 0.1 mg/ml (fibrous structure is lost); on the right, the
concentration is 1 mg/ml.
The formation of the nanofibres and the aggregated structures is not simultaneous. It
was observed that there are three phases of calcium ion uptake: 6 hours, 36 hours and 72
hours from the start of the reaction. The first detectable structures appear after around
fifteen hours. Bundles and cones appear later, in the last phase of the experiment. Also,
some calcite was crystallized at the bottom at the earliest stage of the experiment. These
crystals were found attached to the vessel. Some nanofibres precipitated to the bottom,
unattached to the vessel. (See Figure 4, left picture.) This suggest that the nanoparticles
Control of Calcium Carbonate Crystallization by a Serum Protein
43
Figure 4 Individual fibres and an example of the bundling of fibres. The concentration
of protein is 1 mg/ml.
were formed and assembled in the intermediate phase of the experiment. The remaining
fibres fell down after drying.
We visualise cones as hanging vertically downwards from an upper point. The viscosity
of the aqueous solution had a role to play as well. If we take a vertical transversal cut of one
of the conical shapes, the line that defines the exterior of the crystal will not be straight in
general; its concavity depends on the viscosity. Namely, as the viscosity increases, the cones
appear to be more ‘spread out’ and have a larger angle at the tip. The shape of the bundles
follows a similar pattern. However, the changes in viscosity did not alter the general shape
of the aggregated structures, nor their global behaviours. See Figure 5.
Figure 5 Changes in the opening angle of the fibre cones as a function of the viscosity of
the aqueous solution.
The position of the water surface turned out to be important as well. A variation of
the standard experiment was done in a very thin pipette, which was small enough that,
if turned upside down, the water surface is preserved by surface tension, and the solution
does not drop down. Two pipettes with the solution of protein and CaCl2 were placed in
the desiccator, one upside down and the other upright. The first did not show growth of
crystals, while the other did. Other variations, such as rocking the containers, resulted in
44
Control of Calcium Carbonate Crystallization by a Serum Protein
the absence of crystals as well. This could simply be due to the fragility of the crystal
structures.
This biomineralization-like process starts out with the diffusion of carbon dioxide into
the protein solution. Since no other material is getting in or out the vessel, the time
scales of the uptakes of the structures described above imply that they depend only on the
concentration of each component. Therefore understanding these concentrations is one of
the keys to understanding the interactions with respect to the crystallization process. This
aggregation process is independent of the way the nanoparticles and the fibres are bound
together.
Based on these observations, many interesting phenomena can be subjected to modeling.
For instance, as observed before, The mechanism responsible for the crystallization must
be responsible for the banding structure and the way clusters are formed, as well as the
assembly of the self-similar structures in layers.
Our objective is to model the formation of calcium carbonate, with and without the
protein. Then, considering the appropriate terms in the system, we simulate the uptakes of
nanoparticles, the fibres and finally the crystal growth.
3 Mathematical modelling
3.1 Proposed mechanism. In this section, we propose a mechanism for the formation
of the crystals. The two systems of pdes that we present below describe this process. The
key stages in the process are as follows:
1. Carbon dioxide dissolves into solution of water with calcium chloride and serum
protein.
2. Carbonate ions combine with calcium ions to form calcium carbonate (which is insoluble).
3. Calcium carbonate, in the absence of a serum protein, forms large crystals, that is,
the growth of crystals is not limited.
4. Calcium carbonate, in the presence of the serum protein, form nanoparticles. Though
we do not model their shape or configuration, it is assumed that their morphology
makes the nanoparticles bind into rather large filaments [9, 10]. This indicates some
sort of growth-limiting effect of the serum protein.
5. When the concentration of nanoparticles reaches a critical level, they aggregate, or
self-assemble, to form fibres. In the experiments, the serum protein was never fully
consumed.
6. The fibres assemble to form sheets, bundles and cones. It is not clear whether this
occurs after fibre-formation or whether the two processes occur simultaneously.
Our system of equations describes the three time stages of the problem, namely the
diffusion of calcium and carbonate ions into the solution, the nucleation and growth of
CaCO3 crystals which may occur with interaction with the serum protein and, finally, the
arrangement of the nanoparticles into crystallized superstructures.
3.2 Calcium carbonate formation and nanoparticles. We construct a system of
pdes that are one-dimensional in space, with the z-coordinate measuring depth from the top
of the vessel; thus z = 0 corresponds to the top of the vessel. Here u and v are respectively
the concentrations of calcium ions and carbonate ions at depth z and time t.
The initial conditions are as follows: first, there is no flow of any element or compound
through the bottom of the vessel z = L. In addition, at the top of the vessel, we assume
Control of Calcium Carbonate Crystallization by a Serum Protein
45
Table 1 Summary of parameter values.
Parameter Description
L
u0
vair
h
α
P0
a
k1
k2
k3
k3
k4
ε
D1
D2
D4
D5
Value
Depth of container
Initial concentration of calcium in vessel
Carbon dioxide concentration in air above vessel
Constant for dissolution of CO2(g)
Stoichiometic constant for crystal-protein ratio
Initial concentration of serum protein
Inverse of diameter of nanoparticle Xp
Reaction rate for Ca+CO3 → CaCO3
Reaction rate for αX+P→ Xp
Rate of fibre growth in model system 1
Rate of fibre growth in model system 2
Rate of sidebranching in model system 2
Rate of new fibre nucleation in model system 2
Diffusion constant for calcium ions, u
Diffusion constant for carbonate ions, v
Diffusion constant for microcrystals, X
Diffusion constant for nanoparticles, Xp
6.75 mm
50 ppm
387 ppm
2 × 10−5 mol m−2 s−1
3
1 mg ml−1
1 nm
1 × 10−5
1 × 10−2
1
2 × 10−5
2 × 10−4
2 × 10−3
2 cm2 s−1
2 cm2 s−1
1 cm2 s−1
1 cm2 s−1
a Robin condition for the diffusion of CO2(g) , with no carbon dioxide in the solution at
time t = 0 and a positive atmospheric concentration, vair . We will also assume uniform
concentrations of calcium and protein initially in the solution. Hence we have
u(z, 0) = u0 ,
uz (0, t) = 0,
vz (0, t) = h(vair − v(0, t)),
(3.1)
v(z, 0) = 0,
uz (L, t) = 0,
vz (L, t) = 0
(3.2)
with the constant of dissolution of CO2(g) given in Table 1.
We apply the law of mass action to the governing chemical equations (2.5), which can
k
1
X, to obtain the following system of coupled non-linear
be written chemically as u + v −→
equations
∂v
∂2u
∂2v
∂u
= D1 2 − k1 uv,
= D2 2 − k1 uv,
(3.3)
∂t
∂z
∂t
∂z
where D1 and D2 are the diffusion coefficients for calcium and carbonate ions respectively,
and k1 is the reaction term for the calcium and carbonate ions to combine and form the
insoluble crystal, X, whose concentration we denote by X(z, t).
Denoting the protein concentration by P (z, t), the boundary and initial conditions for X
and P describe the fact that neither can escape from the top or bottom of the reaction vessel.
Whilst there is no crystal present at t = 0, the protein is present, at some concentration,
P0 distributed uniformly, hence we have
Xz (0, t) = 0,
Xz (L, t) = 0,
X(z, 0) = 0,
(3.4)
Pz (0, t) = 0,
Pz (L, t) = 0,
P (z, 0) = P0 .
(3.5)
The second set of pdes describe the formation of CaCO3 when they interact with the serum
protein, which is present at concentration P (z, t). The chemical model we wish to describe
46
Control of Calcium Carbonate Crystallization by a Serum Protein
k
2
is αX + P −→
Xp , where Xp denotes a nanocrystal of calcium carbonate which is large
enough to have adsorbed at least one protein molecule onto its surface. Hence,
∂X
∂2P
∂2X
∂P
= D3 2 − k2 X α P,
= D4 2 − k2 X α P + k1 uv,
(3.6)
∂t
∂z
∂t
∂z
where D3 and D4 are the diffusion coefficients, k2 is the combined reaction term for the
nucleation and growth of a calcium carbonate crystal and its subsequent combining with
protein. Rather than go to the complexity of modelling clusters of the full range of sizes,
we go straight to an approximation in which many particles are assumed to interact to form
a much larger complicated structure. This can be derived in a more rigorous fashion, as
analysed in other work [3, 5], and has been used successfully in a number of applications
from surfactant flow [1, 2] to other models of nucleation, competition and inhibition [4, 11].
The third part of the model has been approached in two different ways, these are
presented in the next two subsections. The first model’s growth occurs only at the tips,
whilst the second takes into account the possibility that fibres may branch as they grow,
giving rise to triangular surfaces.
3.3 Formation of fibre nanocrystals and fibres: system 1. We assume that fibres
are initiated at the top surface and grow vertically downwards, so a fibre of length q ends
at a depth z = q. Let Xp be the concentration of nanoparticles and Y (z, t) the number of
fibres of length z ending at depth z at time t. We assume that nanoparticles can move by
k
diffusion, but that fibres are fixed. Nanoparticles are created by the process αX + P →2 Xp
discussed above, hence Xp is determined by an equation similar to (3.6):
∂ 2 Xp
∂Xp
= D5
+ k2 X α P − k3 aXp Y,
(3.7)
∂t
∂z 2
where D5 is the diffusion coefficient of the nanoparticles Xp , k2 is as before, and k3 is the
reaction term for the nanoparticles and calcium carbonate fibres. The derivation of an
equation for Y (z, t) is more complex.
The quantity Y (z, t) can only change if there are fibres that end at z − δ (for some
small δ 1). Because fibres grow downwards, we must have Y (z − δ) > Y (z). In a
discrete approximation to the system, we let Y (nΔz, kΔt) be the number of fibres of length
nΔz at time kΔt. In the next time interval, a particle Xp will be added with probability
which depends linearly on the concentration Xp , hence we denote this probability by pXp .
Incorporation of tips of length nΔz is done at the expense of tips located at (n − 1)Δz.
Thus we have
Y (nΔz, (k+1)Δt) = Y (nΔz, kΔt) − pXp Y (nΔz, kΔt) + pXp Y ((n−1)Δz, kΔt).
(3.8)
Taking the limits Δz → 0, Δt → 0 with Δz ∼ Δt, and defining k3 = p limΔt→0 (Δz/Δt),
we obtain
∂Y
∂Y
= −k3 Xp
.
(3.9)
∂t
∂z
Hence fibres grow at the rate k3 Xp , with k3 having units of [k3 ] = length per concentration
per time. For (3.7) we require the units of k3 aXp Y to match that of ∂Xp /∂t. Therefore the
units of the constant ‘a’ must be inverse length. In fact 1/a can be thought to correspond
to the diameter of an Xp nanoparticle.
In this system, fibres grow only at the tip, that is, there is no side-branching. In this
formulation, k3 is the reaction term for the nanoparticles and the tips’ aggregated structure.
Control of Calcium Carbonate Crystallization by a Serum Protein
47
3.4 Formation of fibre nanocrystals and fibres: system 2. In this formulation,
as well as fibres growing from the surface downwards with growth only occurring at their
lower ends, we permit the formation of new fibres anywhere in the vessel where there
are sufficient nanoparticles (Xp ), and we allow growing fibres to sidebranch, so that the
aggregated structures become wider at the bottom than the top.
Let Xp (z, t) be as before, but now Y (z, t) is the mass of fibres at depth z; we assume
that two nanoparticles colliding can initiate the growth of a new fibre – which will be a small
effect, hence the rate constant is taken to be ε. More important is the extensional growth
of existing fibres, which occurs with a rate constant k3 ([
k3 ] = length per concentration per
k4 ] = per concentration per time).
time) and the side-branching which occurs at a rate k4 ([
Side-branching causes the crystals to become wider further down the vessel. In accordance
with (3.7) and (3.9) one has in this case
∂ 2 Xp
∂Xp
= D5
+ k2 X α P − k3 aXp Y − εXp2 − k4 Xp Y,
(3.10)
∂t
∂z 2
∂Y
∂Y
= −
k3 Xp
+ εXp2 − k4 Xp Y,
(3.11)
∂t
∂z
where the first term of the latter equation stands, as before, for the growth of the fibres.
k3 a. The boundary conditions for (3.10)–(3.11) are all zero
Typically we expect ε < k4 < flux and the initial conditions are also zero, namely
Xp,z (0, t) = 0,
Xp,z (L, t) = 0,
Xp (z, 0) = 0,
(3.12)
Yz (0, t) = 0,
Yz (L, t) = 0,
Y (z, 0) = 0.
(3.13)
3.5 Diffusion constants and other observations. Now we face the problem of
estimating the constants for the various processes. These rates are known for the formation
of calcium carbonate in alkaline environments, but not for environments where the serum
protein is present; in these cases they have to be inferred. Rate constants are extracted
from time dependent concentration profiles determined from experiment. Such profiles are
depicted in Figure 6.
In the case of α, we set a value of 3, instead of 300, since for large α the equations
become too stiff for the numerical method used. However, we can infer the results for large
α. This will be detailed below. We also estimated the rest of the unknown coefficients,
using similar principles.
4 Results
The systems of pdes were solved using MatLab 7.5. The graphs below show the concentration of each component in the time-space phase plane. The color scale goes from blue to
red, where blue signifies the absence of the component, and red the highest concentration.
4.1 Formation of fibre nanocrystals and fibres: system 1. Here we solve equations (3.1)–(3.9). First we present the solution of the system in the absence of protein, see
Figure 7. The concentrations of calcium, carbonate ions, and calcium carbonate crystals
conform to experimental observation. That is, nearly uniform production of calcium carbonate microcrystals with carbonate diffusing into solution from the position z = 0. We
now compare Figures 7 and 8, the latter having protein in the solution. In both cases,
the concentration of calcium decays through the experiment but is always nearly uniform
through the vessel, the concentration of carbonate starts at zero and increases, initially
48
Control of Calcium Carbonate Crystallization by a Serum Protein
Figure 6 This graph shows the concentration of Ca2+ in the solution plotted against time
for the cases with (red) and without (blue) protein. Without protein the concentration
falls very slowly whereas with protein there is a significant decrease at about t = 10. The
green data points are more careful measurements which compensate for the build up of
CaCO3 on the probes. These values are acquired less frequently (isolated data at t =
10, 25, 50).
from the top of the vessel, and always has a significant gradient from top to bottom of the
vessel. The concentration of calcium carbonate has similar behaviour, although with a less
pronounced gradient.
In Figure 8, we see the concentration of nanocrystals, fibres and the consumption of
the protein in accordance with the first system of pdes, where the initial concentration
of the protein is not zero. In this simulation, the protein is consumed uniformly, but the
nanocrystals at the top of the vessel are used first in the assembly of aggregated structures
(fibres). In this case the assembly of nanocrystals into fibres occurs so rapidly that we
see the formation of fibres from the top of the vessel, and never observe any significant
concentration of nanocrystals, except for midway through the simulation when there is a
small concentration at the bottom of the vessel. They cannot attach to fibres, because at
this point in the simulation there are only fibres at the top of the vessel. Note that fibres
grow monotonically.
Now we test the results of the simulation by varying the concentration of protein, and
find results consistent with the experimental observations described above. At lower protein
concentrations, it will take longer for the crystals (X) to interact with protein forming the
nanoparticles Xp . Hence the formation of fibres would also be delayed; see Figure 9 top. An
additional process in the experiment which has yet to be built into our models is that the
crystals would grow larger before having their growth inhibited by the protein, and hence
there would be fewer, but larger nanoparticles to form the fibres. On the other hand, if the
Control of Calcium Carbonate Crystallization by a Serum Protein
49
Figure 7 Solution of (3.1)–(3.9) without the addition of any protein. In this situation
calcium chloride forms throughout the length 0 < z < L.
concentration of protein is higher, the increase in fibre density occurs sooner, due to the
more rapid formation of the Xp nanoparticles; see Figure 9 bottom.
As pointed out above, we have set α to 3, instead of 300, since for large α the equations
become too stiff for the numerical method used. This parameter, determines the size of
the nanoparticle. If its value increases, more calcium carbonate would be needed to form
the aggregated structures. However, the global behaviour of the system would remain
unaffected, and differences in timescales caused by changing α can be compensated for by
simultaneously changing k2 .
4.2 Formation of fibre nanocrystals and fibres: system 2. Here we solve equations (3.1)–(3.6) and (3.10)-(3.13). Considering the possibility of branching allows us to
describe other characteristics that we could not see using the first system. For a small
amount of side branching the concentration profiles of the various species are not suprisingly quite similar to Figure 8 and are omitted for brevity. When nucleation of new fibres
is significant in the model, we obtain a much clearer front in the formation of fibres as
evidenced in Figure 10.
4.3 Effect of variations in viscosity. Recall that the viscosity of the aqueous solution did not alter the growth of the fibres, but did change the shape of the conical aggregate
50
Control of Calcium Carbonate Crystallization by a Serum Protein
Figure 8 Solution of (3.1)–(3.9) with the addition of a small amount of serum protein.
Notice that the protein depletes uniformly in space. Nanocrystals form throughout but
are incorporated into fibres starting at the top of the vessel.
(which we are not concerned with in this model). Since our model has no viscosity parameter, we model the change by changing the diffusion constants, and noting that these are
given by D = kB T /6πaη, where kB is Boltzmann’s constant, T is temperature, a is the radius of the diffusing particle and η is the viscosity. Hence a doubling viscosity corresponds
to reducing all the diffusion constants by a factor of two. Changing the viscosity had no
appreciable effect for system 1 or system 2 with small side branching. However for system 2
with significant side branching, the crystallization front of both the nanoparticles and fibres
is smoothed out by this change in viscosity; see Figure 11.
5 Discussion
We have formulated a model for the concentrations of the various species of ion, protein,
microcrystals, and larger scale crystal complexes present in the system as a function of depth
in the vessel, which describes their variation through the time course of the experiment. The
models have the form of a coupled system of partial differential equations with parameters
which we have derived from crude calculations based on experimental data. The ordering
of events, and predicted form of the solution matches well with the observed data, although
the model lacks the spatial resolution to describe the precise geometry of structures formed.
We speculate that this structure may be related to the properties of the actual molecular
level details of the protein present in solution.
Control of Calcium Carbonate Crystallization by a Serum Protein
51
Figure 9 The effect of varying the initial protein concentration. Above the initial protein
is 0.5 units, and below the initial protein is 2 units. Compare these with Figure 8, where
the concentration is 1 unit.
We predict the presence of a diffusive wave of calcium carbonate which travels down
through the reaction vessel. Serum protein interacts with these particles to form micocrystals coated in protein that then self-assemble into bundles of fibres which occasionally
undergo side-branching. Side-branching forces the growing surface to become wider the
further down the vessel it grows, causing an inverted cone-shaped structure to form.
Our model does not capture the geometry of the described aggregation phenomena
beyond the fibres; namely, the ensemble of bundles and cones is not described. However,
we can derive some conjectures that are consistent with the models described above, and
the experimental data. For instance, at moderate concentrations of protein, fibres should
grow in bundles, with a distribution of lengths. At intermediate concentrations, the fibres
should grow to a length where secondary effects (competition of nucleation vs. increasing
electrostatic energy) inhibit further growth. Under these conditions, cones will appear.
Finally, for large concentrations of protein, nucleation should proceed on the inner surfaces
of cones, and banding should appear. There are thus many questions which this report
leaves open for further study.
52
Control of Calcium Carbonate Crystallization by a Serum Protein
Figure 10 Solution to (3.1)–(3.6) and (3.10)-(3.13) when there is a significant amount of
side branching. Compare to Figure 8 where no side branching occurs.
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53
Figure 11 Solution to (3.1)–(3.6) and (3.10)-(3.13) with a solution of twice the viscosity
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